Q8_1130 - MATH 415.01 QUIZ 8A Mar 3 2010 SHOW ALL YOUR\VORK...

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Unformatted text preview: MATH 415.01 QUIZ 8A Mar 3, 2010 SHOW ALL YOUR \VORK! GOOD LUCK! NAME: 1. (6.5 pts.) Consider the conduction of heat in a rod 40 em in length Whose ends are maintained at 0°C for all t > 0. Suppose that a2 = 1. Find an expression for the temperature u.(:r, t) if the initial temperature distribution in the rod is . " H 1“ l ; ,.‘ .7_» (a: 0) 55: 0 S I < 20, " \Ur'rl :3 "iv—a “'i -i U z , i 40 — (E, 20 S .'L‘ < 40 l I“; 1 ;_ :2 ' s i ' 5. “ t ‘- ._ 7,;, _.j i \ Rx ‘ ‘ ll _ .—‘. l V ..-M 3' L I r . \ '\ 1 r s— X _. l n L, 7"“ c; 3 4.» J \ 4 * ~_\ ‘ “'r"l/ ‘3.”4' ‘1‘ h D _ .1'} .1 . ‘ I \ » . ‘ \‘x \ er" I- a. -. l . \ J \'\ \ , W’\ ' " ‘ _: 3'51" ‘ i. —~ I. J V» t __ c; a.) .1- 3 2 ' v ‘ A'- l «v— ‘ :j {j 2. (6 pts.) Let an aluminum rod of length 20 cm be initiall}r at the uniform temperature of 25°C. Suppose that at time t = 0 the end a: = 0 is cooled to 0°C while the end LII = 20 is heated to 60°C, and both are thereafter maintained at those temperatures. Find the steady state (final) temperature distribution. Then , sketch the graph of the initial temperature distribution and final temperature distribution on the same set of axes. ‘ If. H a. u - .r- I a" If; \ I; .7; _.:I_,\_,. W». s- o r- \t an 4 L ~ w- . (i. '\-.‘ \ \gi l I h {J ‘ ’ 'I i . LA 1 s- 3:: x.»- 3,: \‘ I \J'lx 5,1; L. l r- . ‘ w" . _ r .. o \‘h L -- r ,- ‘ éfif—fidir- 3.: I, f/"' — :' _____ i is” g 7 ii 'l._/ MATH 415.01 QUIZ 8B Mar 3, 2010 SHOW ALL YOUR WORK! GOOD LUCK! NAME: 1. (5 pts.) Determine whether the method of separation of variables can be used to replace the given partial differential equation by a pair of ordinary differential equations. If so, find the equations. Otherwise, explain why? umr'l’umt‘l‘ut :0 H: X(x)T(+\ XML X'T'flfTL/J X“? :“Ti (KITX) X" , 3' _ 2. (7 .5 pts.) Consider a bar 30 cm long that is made of a material for which {1-2 = 1 and whose ends are insulated. Suppose that the initial temperature is zero except for the interval 5 < a: < 10, where the initial temperature is 25°. Find the temperature 12(55, t). LAK)‘ ': ‘Jl —l_ ~r- ’ “ _ . . L i :3 l “1 ex (Qua) 4 (UT/Hi 0 fit? 0 A63" UQK'C’WLé 15 “X4” ‘ (3 1:4xé3j 3&— T — ‘>@ \ +AT :Q 99 Ala.” "‘ “Eco - we erase 0 ,3 J Li”): A {if L m. ‘_ 1 'L L rib—ML so K1 _ r, J” , _).5 ::HK4’(2__ X" le’M‘: fl- Clwktfl‘ll' a ‘- ‘prrl'x : 3-3:; 5/15 AX- é X 1: Lt ’0 4(23‘;.\J(Ay {CLng‘lmu’ ‘ 0 ~ $13 5 154-0 mm W050 1”“ ' m a ll 3C2. bvm&$ (£3130 :3 Clio n of; ax:5_o kmrfii’ (—MA‘V‘J“ JD 2'7- Xllkl :0 3? CLEO Cn-i§ 2—500} A“ Slaw) HAM _ r r -- '1‘ g b - K : (3‘b_,y\x l a Ibrllifir "‘ hir- ' I — 2.x?! e 12 4:3 ...
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Q8_1130 - MATH 415.01 QUIZ 8A Mar 3 2010 SHOW ALL YOUR\VORK...

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